Originally posted by sonhouse
Just that it was the first one found and to be a 5 digit # like finding a needle in a haystack although he had to think it through very well indeed. I was just thinking out of all the numbers up to 78K, there could well have been others smaller, thats all.
John Selfridge never proved that k = 78557 was the smallest Sierpinski number. If he had then we wouldn't have the Sierpinski problem. Up until the start of the Seventeen or Bust project it had been shown for all but 17 values of k under 78557 there existed a prime number of the form k * 2^n + 1. The seventeen or bust project concentrates on the remaining numbers and attempts to find a prime number for each remaining k value. If they suceed then they will have proven (via brute force) that 78557 is the smallest Sierpinski number.
Selfridge made his discovery by a proof rather than brute force (ie showed that no prime number can be of the form 78557 * 2^n + 1). He likely suspected no smaller value existed but obviously the enormity of the task didn't make it possible to prove it.
And of course there could well have been values smaller, that's the case for everything. The smallest number that is the sum of two cubes two different ways is 1729. You may as well ask why isn't there a smaller number than that? There just isn't.